Problem: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{p^2 - 49}{p + 7}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = p$ $ b = \sqrt{49} = 7$ So we can rewrite the expression as: $y = \dfrac{({p} + {7})({p} {-7})} {p + 7} $ We can divide the numerator and denominator by $(p + 7)$ on condition that $p \neq -7$ Therefore $y = p - 7; p \neq -7$